LCM and HCF

LCM (Least Common Multiple): The smallest multiple that is exactly divisible by two or more numbers.

HCF (Highest Common Factor): The largest factor that divides two or more numbers without leaving a remainder.

Two big fancy names but don’t be put off - they’re both real easy. There are two methods for finding each - this page starts you off with the nice, straightforward methods.

METHOD: 1) List the multiples of all the numbers. Find the smallest one that’s in all the lists.

The LCM is sometimes called the Least (instead of “Lowest”) Common Multiple.

EXAMPLE

Find the lowest common multiple of 12 and 15.

To find the multiples:

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, …

Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, …

The smallest multiple that appears in both lists is 60, so the LCM of 12 and 15 is 60.

METHOD: HCF

To find the Highest Common Factor (HCF) of two numbers, identify the largest number that exactly divides both values without leaving a remainder.

For example, to determine the HCF of 12 and 15, we can list their factors:

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 15: 1, 3, 5, 15

The common factors are 1 and 3, therefore the HCF is 3.

EXAMPLE

Find the highest common factor of 36, 54 and 72.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36; the factors of 54 are 1, 2, 3, 6, 9, 18, 27, 54; and the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The common factors are 1, 2, 3, 6, 9, and 18, so the HCF is 18.

Just take care listing the factors - make sure you use the proper method (as shown on the previous page) or you’ll miss one and blow the whole thing out of the water.

LCM and HCF - learn what the names mean

Q1) Find the LCM of 12, 14 and 21. To find the LCM, we first list the prime factors of each number: 12 = 2^2 × 3, 14 = 2 × 7, and 21 = 3 × 7. Next, we take the highest power of each prime that appears in the factorization: 2^2, 3^1, and 7^1. Multiplying these together, we get LCM = 2^2 × 3^1 × 7^1 = 4 × 3 × 7 = 84.

Q2) Find the HCF of 36 and 84. To find the HCF, we first list the prime factors of each number: 36 = 2^2 × 3^2 and 84 = 2^2 × 3^1 × 7^1. Then, we take the lowest power of each common prime factor: 2^2 and 3^1. Multiplying these together, we get HCF = 2^2 × 3^1 = 4 × 3 = 12.